Question 25475: Inclusion-Minimal Covers Need Not Have Minimum Cardinality

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0049-25475 · q-20261015-advance-20261114-r07-vertex-cover-0100

Original question and givens

Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, c1, l1_0, l1_1, l1_2], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c1, l1_0], [c1, l1_1], [c1, l1_2]] and unique candidate subset=[l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

G001 — Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, c1, l1_0, l1_1, l1_2], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c1, l1_0], [c1, l1_1], [c1, l1_2]] and unique candidate subset=[l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

mko-vertex-cover

Step-by-step derivation

S001 · implies

Before

source roles = [vcover-minimal-vs-minimum-boundary]

After

derived = [[candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate]]

Conditions: none

List the full source field order without JavaScript.LP boundary modes all have6 fields but select different roles from the original mode,not answer shape.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [[c0, l0_0, l0_1, l0_2, l0_3, c1, l1_0, l1_1, l1_2], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c1, l1_0], [c1, l1_1], [c1, l1_2]], [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2]]

After

derived = [[c0, c1, l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c1, l1_0], [c1, l1_1], [c1, l1_2]], [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2]]

Conditions: none

The carrier is nonempty with at most12 vertices.Normalize endpoints by JS UTF16<,then source English canonical edge order.Reject loops/reversed duplicate edges;this is not a directed graph or edge cover.

Step identifiers and dependencies

Rule: Vcover_complete_simple_undirected_graph_UTF16_then_English_order · Uses: G001, S001

S003 · implies

Before

source roles = [0, [c0, l0_0], [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2]]

After

derived = [[No, Yes], Yes]

Conditions: none

This edge is covered iff at least one endpoint is selected.Check the entire edge table,not vertex count.

Step identifiers and dependencies

Rule: Vcover_EVERY_edge_selected_endpoint_disjunction · Uses: G001, S002

S004 · implies

Before

source roles = [1, [c0, l0_1], [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2]]

After

derived = [[No, Yes], Yes]

Conditions: none

This edge is covered iff at least one endpoint is selected.Check the entire edge table,not vertex count.

Step identifiers and dependencies

Rule: Vcover_EVERY_edge_selected_endpoint_disjunction · Uses: G001, S003

S005 · implies

Before

source roles = [2, [c0, l0_2], [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2]]

After

derived = [[No, Yes], Yes]

Conditions: none

This edge is covered iff at least one endpoint is selected.Check the entire edge table,not vertex count.

Step identifiers and dependencies

Rule: Vcover_EVERY_edge_selected_endpoint_disjunction · Uses: G001, S004

S006 · implies

Before

source roles = [3, [c0, l0_3], [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2]]

After

derived = [[No, Yes], Yes]

Conditions: none

This edge is covered iff at least one endpoint is selected.Check the entire edge table,not vertex count.

Step identifiers and dependencies

Rule: Vcover_EVERY_edge_selected_endpoint_disjunction · Uses: G001, S005

S007 · implies

Before

source roles = [4, [c1, l1_0], [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2]]

After

derived = [[No, Yes], Yes]

Conditions: none

This edge is covered iff at least one endpoint is selected.Check the entire edge table,not vertex count.

Step identifiers and dependencies

Rule: Vcover_EVERY_edge_selected_endpoint_disjunction · Uses: G001, S006

S008 · implies

Before

source roles = [5, [c1, l1_1], [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2]]

After

derived = [[No, Yes], Yes]

Conditions: none

This edge is covered iff at least one endpoint is selected.Check the entire edge table,not vertex count.

Step identifiers and dependencies

Rule: Vcover_EVERY_edge_selected_endpoint_disjunction · Uses: G001, S007

S009 · implies

Before

source roles = [6, [c1, l1_2], [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2]]

After

derived = [[No, Yes], Yes]

Conditions: none

This edge is covered iff at least one endpoint is selected.Check the entire edge table,not vertex count.

Step identifiers and dependencies

Rule: Vcover_EVERY_edge_selected_endpoint_disjunction · Uses: G001, S008

S010 · implies

Before

source roles = [[[[c0, l0_0], Yes], [[c0, l0_1], Yes], [[c0, l0_2], Yes], [[c0, l0_3], Yes], [[c1, l1_0], Yes], [[c1, l1_1], Yes], [[c1, l1_2], Yes]]]

After

derived = [Yes, null, []]

Conditions: none

Conjoin all edges and retain the first uncovered edge in source order,or null,notFalse.Isolated vertices have no incident coverage obligation and are optional.

Step identifiers and dependencies

Rule: Vcover_COMPLETE_edge_conjunction_FIRST_uncovered_or_NULL · Uses: G001, S009

S011 · implies

Before

source roles = [l0_0, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2]]

After

derived = [[l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], [[[c0, l0_0], No], [[c0, l0_1], Yes], [[c0, l0_2], Yes], [[c0, l0_3], Yes], [[c1, l1_0], Yes], [[c1, l1_1], Yes], [[c1, l1_2], Yes]], No]

Conditions: none

Delete only this vertex and recheck every edge,without reusing another deletion result.

Step identifiers and dependencies

Rule: Vcover_EACH_single_vertex_deletion_complete_edge_test · Uses: G001, S010

S012 · implies

Before

source roles = [l0_1, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2]]

After

derived = [[l0_0, l0_2, l0_3, l1_0, l1_1, l1_2], [[[c0, l0_0], Yes], [[c0, l0_1], No], [[c0, l0_2], Yes], [[c0, l0_3], Yes], [[c1, l1_0], Yes], [[c1, l1_1], Yes], [[c1, l1_2], Yes]], No]

Conditions: none

Delete only this vertex and recheck every edge,without reusing another deletion result.

Step identifiers and dependencies

Rule: Vcover_EACH_single_vertex_deletion_complete_edge_test · Uses: G001, S011

S013 · implies

Before

source roles = [l0_2, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2]]

After

derived = [[l0_0, l0_1, l0_3, l1_0, l1_1, l1_2], [[[c0, l0_0], Yes], [[c0, l0_1], Yes], [[c0, l0_2], No], [[c0, l0_3], Yes], [[c1, l1_0], Yes], [[c1, l1_1], Yes], [[c1, l1_2], Yes]], No]

Conditions: none

Delete only this vertex and recheck every edge,without reusing another deletion result.

Step identifiers and dependencies

Rule: Vcover_EACH_single_vertex_deletion_complete_edge_test · Uses: G001, S012

S014 · implies

Before

source roles = [l0_3, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2]]

After

derived = [[l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], [[[c0, l0_0], Yes], [[c0, l0_1], Yes], [[c0, l0_2], Yes], [[c0, l0_3], No], [[c1, l1_0], Yes], [[c1, l1_1], Yes], [[c1, l1_2], Yes]], No]

Conditions: none

Delete only this vertex and recheck every edge,without reusing another deletion result.

Step identifiers and dependencies

Rule: Vcover_EACH_single_vertex_deletion_complete_edge_test · Uses: G001, S013

S015 · implies

Before

source roles = [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2]]

After

derived = [[l0_0, l0_1, l0_2, l0_3, l1_1, l1_2], [[[c0, l0_0], Yes], [[c0, l0_1], Yes], [[c0, l0_2], Yes], [[c0, l0_3], Yes], [[c1, l1_0], No], [[c1, l1_1], Yes], [[c1, l1_2], Yes]], No]

Conditions: none

Delete only this vertex and recheck every edge,without reusing another deletion result.

Step identifiers and dependencies

Rule: Vcover_EACH_single_vertex_deletion_complete_edge_test · Uses: G001, S014

S016 · implies

Before

source roles = [l1_1, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2]]

After

derived = [[l0_0, l0_1, l0_2, l0_3, l1_0, l1_2], [[[c0, l0_0], Yes], [[c0, l0_1], Yes], [[c0, l0_2], Yes], [[c0, l0_3], Yes], [[c1, l1_0], Yes], [[c1, l1_1], No], [[c1, l1_2], Yes]], No]

Conditions: none

Delete only this vertex and recheck every edge,without reusing another deletion result.

Step identifiers and dependencies

Rule: Vcover_EACH_single_vertex_deletion_complete_edge_test · Uses: G001, S015

S017 · implies

Before

source roles = [l1_2, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2]]

After

derived = [[l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], [[[c0, l0_0], Yes], [[c0, l0_1], Yes], [[c0, l0_2], Yes], [[c0, l0_3], Yes], [[c1, l1_0], Yes], [[c1, l1_1], Yes], [[c1, l1_2], No]], No]

Conditions: none

Delete only this vertex and recheck every edge,without reusing another deletion result.

Step identifiers and dependencies

Rule: Vcover_EACH_single_vertex_deletion_complete_edge_test · Uses: G001, S016

S018 · implies

Before

source roles = [[c0, c1, l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c1, l1_0], [c1, l1_1], [c1, l1_2]], [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], No], [l0_1, [l0_0, l0_2, l0_3, l1_0, l1_1, l1_2], No], [l0_2, [l0_0, l0_1, l0_3, l1_0, l1_1, l1_2], No], [l0_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], No], [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_1, l1_2], No], [l1_1, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_2], No], [l1_2, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], No]]]

After

derived = [512, [0, 0, 1, 7, 22, 40, 44, 29, 9, 1], [3, [c0, c1]], Yes, 2, No, No]

Conditions: none

Count covers by size over all2^n subsets.Zero counts at smaller sizes give a lower bound and the explicit witness attains it.By superset monotonicity,any proper-subset cover would make some single deletion a cover,so deletion tests characterize inclusion-minimality,not minimum cardinality.The last flag compares two Booleans;bothFalse may agree.

Step identifiers and dependencies

Rule: Vcover_ALL_subsets_cardinality_counts_minimum_witness_and_GENERAL_minimality · Uses: G001, S017

Reverse checks

S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], No], [l0_1, [l0_0, l0_2, l0_3, l1_0, l1_1, l1_2], No], [l0_2, [l0_0, l0_1, l0_3, l1_0, l1_1, l1_2], No], [l0_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], No], [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_1, l1_2], No], [l1_1, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_2], No], [l1_2, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts

input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: e74e40f53d9f973b9b14ab8444eab153d4097f39dd0d9a2852dec8a28c1eeb1d
candidate record SHA-256: dc057917cec614b2dad18085176a00e796b81d29b8b11b9dda2a9d557a3517e9
semantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3
question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3
Lean: NOT_RUN · human review: NOT_PERFORMED

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